Godzilla also absorbed electricity and became even stronger in Godzilla vs. Mechagodzilla. Both monsters have been shown to thrive off of electricity at some point. While there have been a handful of occasions where humans have used electricity to fend Godzilla off, it was never stated that he was particularly weak against it. Considering how unmoved by the electric current Godzilla was in the original movie, I'd say he's not particularly weak against it.
It would appear that a higher voltage of electricty is what actually hurts him in the films of the early 60's. The first Godzilla plowed through power lines..but that was a different Godzilla.
MECHAGODZILLA '74 was released a decade after MOTHRA VS. GODZILLA, and the powers Godzilla was given in the 70's, were absolutely ridiculous. The only reason it 'charged' him in that one film is because the script called for it...just like the script called for him to (clearly) be weak to electricity in 1962; to give Kong an advantage. Having him be weak to it in a video game for the sake of balance really isn't that big of a deal, IMO, considering it was interchangeable in the films anyway. Like I said earlier, Godzilla needs to be weak to something, because he's already insanely good.
It seems like Alien Energy would be a reasonable weakness for Godzilla.
Yeah, that could work.